Peter Woit has a new post with this title, in which his main point seems to be that the tensor product of two spinors is a complex scalar plus vector, whereas it represents a physically real scalar (energy/time) plus vector (momentum/position). Apparently this is a “mystery”. He has allowed me to post three comments so far, but perhaps I shouldn’t push my luck. My first comment pointed out that if you take the spinors to be quaternions, rather than pairs of complex numbers, then the tensor product of two spinors is in fact real and not complex. No mystery. So he moved the mystery somewhere else, basically saying that you need complex numbers for electromagnetism. Which is true, but it isn’t a mystery.
What is a mystery is why physicists think they can get the complex numbers needed for electromagnetism out of the quaternions that are needed for spin, rather than keeping the complex numbers and quaternions separate. By doing so they lose a lot of information that is necessary for physics, including many properties of mass – such as the three generations of electrons – that are essential for developing a quantum theory of gravity.
Woit points out that the Dirac operator is a vector, although it “looks like” a scalar (so he says). Actually, it is a scalar plus a vector, since he is talking non-relativistically in this discussion. So when you apply the Dirac operator to a spinor you get spins (0+1) x 1/2 = 1/2 + 1/2 + 3/2, or in algebraic language (R + R^3) x H = H + H + H^2. The Dirac equation works with the two spin 1/2 representations here, and throws away the spin 3/2 representation. Which is a pity, because you need the spin 3/2 representation to understand the three generations. You also need it for a chiral theory, because although you can choose the two spin 1/2 representations to be one left-handed and one right-handed, you don’t have that choice with the spin 3/2 representation – either it is left-handed or it is right-handed, and that’s that. You also need it for quantum gravity, because the tensor product of spin 1/2 with spin 3/2 is spin 1 (vector, or gravitational field) plus spin 2 (gravitational waves).
We can’t entirely blame Dirac for throwing away the spin 3/2 representation, because the muon wasn’t discovered until several years after he wrote down his equation. But perhaps someone should have realised at some point that it is actually needed to explain the existence of the muon. As it stands today, no-one has a clue why the muon exists. It’s a mystery, apparently.
No it isn’t. Do the maths properly, for God’s sake. These things are only a mystery if you are careless with the mathematics, and don’t correct your mistakes, even after 100 years. If you try to put all the mathematics of H + H + H^2 into a single Dirac spinor, then of course you have to break the symmetry of the quaternions. If you try and pretend the spinor is just H+H, left-handed plus right-handed spin 1/2, then you lose the mass information, unless you also restrict from H to C and also regard it as C + C + C^2. And even when you do this, you lose the three generations, and have a theory (Dirac algebra) that only deals with one generation at a time. And the theory is a mess.
There is no mystery about any of this. It is sheer bloody mathematical incompetence.